Two laws that want opposite tissue

There is a small canon of empirical relations about cellular tilings, assembled between the 1920s and the 1980s from soap froths, metal grains, and biological epithelia. Two of them are quoted more than the rest, usually in the same paragraph and usually as though they described the same thing.

Lewis’s law relates a cell’s area to its number of sides: bigger cells have more sides.

Aboav’s relation, in the form Weaire gave it, relates a cell’s number of sides to its neighbours’: a cell with many sides is surrounded by cells with few. Written

$$n \, m(n) = (6 - a)n + (6a + \mu_2)$$

where $m(n)$ is the mean side count of the neighbours of an $n$-sided cell and $\mu_2$ is the variance of side counts across the tiling. The parameter $a$ is what gets quoted, and the value usually given is around 1.2.

They are different kinds of statement — one metric, one topological — and there is no obvious reason a tiling should obey both or neither. So it is worth measuring both, on the same material, with the same code.

The measurement #

The same two point sets as the previous essay: nine hundred points at the golden divergence, and nine hundred points thrown at random into the same disc. Voronoi cells, the same rim cut, the same interior population.

Lewis’s law. Golden head: slope 0.009 against his 0.25. Random set: slope 0.231, $n_0 = 1.64$. The random set obeys it; the ordered one does not.

Aboav’s relation. Golden head: $a = 1.18$. Random set: $a = 0.59$. The ordered tiling lands on the value the literature quotes; the random one is half of it.

So each tiling satisfies one law and fails the other, and they fail in opposite directions.

How Aboav’s parameter is fitted #

The relation is usually presented as a line through class means, one point per side count, and fitted that way it is being asked a question with five or six data points.

It is fitted here on the cells instead. Every interior cell contributes a point: its own side count $n$ on one axis, $n \, m(n)$ — its side count times the mean side count of its neighbours — on the other. A least-squares line through those points has slope $6 - a$, and $a$ follows.

Two consequences of doing it that way. The fit is weighted by how common each class is, automatically, which the class-mean version is not: in a golden-angle head the six-sided class holds three quarters of the cells and a class-mean fit would give it the same weight as the seven-sided class holding a tenth.

And the residual scatter is visible, which matters for the same reason it mattered for Lewis’s law. Aboav’s relation is a statement about means, and how much of the variation it explains is a separate question from whether the line fits.

Neighbour sets come from the Delaunay triangulation rather than from the Voronoi polygons, which is the same relation read off the dual — two points are Voronoi neighbours exactly when they are joined by a Delaunay edge. Using the triangulation avoids re-deriving adjacency from polygon geometry, where a shared vertex and a shared edge are easy to confuse.

Why Aboav’s relation likes order #

The mechanism is easier to see than Lewis’s and it is worth spelling out, because it explains why the ordered tiling does well.

In a tiling where the mean side count is forced to six — and it is, by Euler’s formula, which leaves the cells no choice — a cell with seven sides has taken a side from somewhere. Its neighbours are, on average, obliged to have fewer. The relation is a bookkeeping consequence of the constraint, sharpened by the fact that a defect in a nearly regular tiling is local: a seven-sided cell in a golden-angle head sits directly against a five-sided one, almost always, and the pair is the standard defect of a triangular lattice.

That pairing is exactly what makes $a$ come out near its textbook value here. The ordered tiling’s defects are tightly correlated, because they are geometrically obliged to be.

In a random tiling the correlation is weaker. A cell with nine sides is surrounded by cells of every kind, and the compensating deficit is spread over a wider neighbourhood rather than concentrated on the immediate ring. The measured $a$ falls, and 0.59 is what that looks like.

Why Lewis’s law likes disorder #

The previous essay works this out at length, so briefly: Lewis’s law relates variation in area to variation in side count, and the ordered tiling has almost no variation in area to relate. Its cells all have the same size whatever their topology. So the slope goes to zero, not because the relation is violated but because there is nothing for it to describe.

The two laws therefore need opposite things. Aboav’s needs the defects to be local, which happens when the tiling is regular. Lewis’s needs the areas to vary with the topology, which happens when it is not.

What that means for how they are used #

The two are routinely cited together as evidence that some tissue is “a normal cellular structure”. The measurement here says that phrase does not pick out a single class of object.

Concretely: a tissue that satisfies Aboav’s relation with $a \approx 1.2$ is being described as regular, with tightly paired defects. A tissue that satisfies Lewis’s law with $n_0 \approx 2$ is being described as disordered, with a real spread of cell sizes correlated with topology. A paper reporting both, on the same material, is reporting two things that pull in opposite directions — which may be fine, since real tissue sits between the two extremes tested here, but it is not the confirmation it reads as.

The stronger version of the point: each law is a summary statistic devised on the material it was measured on, and neither is a general property of tilings. Aboav worked on metal grain boundaries and soap froths. Lewis worked on cucumber epidermis. Those materials sit at different places on the order–disorder axis and each law is strongest at its author’s end of it.

The five–seven pair #

The defect structure that makes Aboav’s relation work on the ordered tiling deserves a look, since it is a real feature of the pattern rather than a statistical artefact.

A perfectly triangular lattice has every cell six-sided. Wrapping one onto a disc where the density changes with radius makes that impossible, and the mismatch is accommodated by pairs: a five-sided cell adjacent to a seven-sided one. The pair is topologically neutral — the two deficits cancel — so the tiling’s mean stays at six, as Euler’s formula requires.

These pairs sit where the lattice is changing from one parastichy pair to the next, which is to say at the transition radii that the cylinder’s ladder predicts. A seed head’s topological defects are not scattered; they are concentrated in rings, and the rings are at computable radii.

That is why Aboav’s $a$ comes out at its textbook value here. The relation measures how tightly a cell’s excess is compensated by its immediate neighbours, and in this tiling the compensation is as tight as it can be — one cell away, in a bound pair.

It also means the parameter is measuring something quite different in the two cases. On the ordered tiling it measures the locality of a defect structure; on the random one it measures a diffuse statistical correlation. Same number, same formula, two phenomena.

The third summary statistic that goes wrong #

This is now the third time on this site that a single-number summary of a tiling has behaved perversely, and the pattern is consistent enough to be worth naming.

The first was area evenness. The disc essays swept the divergence angle and asked which angle gives the most uniform cell areas, expecting the golden angle. The winner is a rational angle — whose points lie on radial rays, whose sliver cells are near-identical, and between whose rays are enormous empty wedges. The statistic measured what it was defined to measure and that turned out not to be evenness of packing in any useful sense.

The second was Lewis’s slope, which is lowest for the best-packed tiling.

The third is Aboav’s $a$, which is at its textbook value for the tiling that most conspicuously is not the material the relation was measured on.

The common structure: a quantity is defined on disordered material, found to take a characteristic value, and then treated as diagnostic. Applied to ordered material it either collapses, inverts, or coincidentally agrees, and in none of those cases does it mean what the name says. This is not a subtle failure and it is not rare — it is what happens when a statistic outlives the sample it was calibrated on.

What happens between the two extremes #

Real tissue is neither a golden-angle lattice nor a Poisson set, so the interesting question is what both statistics do in between — and the jitter knob on the other essay’s figure answers it for one of them.

Lewis’s slope climbs monotonically with disorder: 0.009 at zero, 0.079 at half a spacing, approaching 0.24 by a full one. There is no threshold and no crossover; the relation simply gets stronger as there is more for it to describe.

Aboav’s $a$ falls over the same range, from 1.18 towards 0.6. It also falls monotonically, and the two curves cross somewhere in the middle.

So there is a disorder level at which a tiling satisfies both relations moderately and neither well, and it is presumably where a good deal of real material sits. That is not a comfortable place to be quoting either law from, and it is the honest reading of a tissue that reports $a = 0.9$ and a Lewis slope of 0.15: not “obeys both”, but “is between the two structures each law was devised for”.

What is asserted #

The build requires both halves of the inversion, and each could fail independently.

The ordered head must satisfy Aboav’s relation with $a$ within 0.35 of 1.2. It does, at 1.18.

The random set must not, and it does not, at 0.59.

Separately, and in the other essay’s check, the random set must satisfy Lewis’s law with $n_0$ within 1.2 of 2 — it does, at 1.64 — and the ordered head’s slope must be less than a quarter of the random one’s, which it is by a factor of twenty-five.

Four assertions, two of which are requirements that a well-known law fail. That is unusual enough to justify: a check that only ever requires agreement with the literature cannot discover that the literature’s scope is narrower than advertised, and the scope is the finding here.

What a well-behaved statistic looks like #

Three summary statistics misbehaving in one collection invites the reading that summary statistics are simply untrustworthy, which is too strong. There is one on this site that behaves, and the contrast says what the difference is.

The largest-gap measurement asks how the biggest empty patch in a head grows as the head fills. For a rational divergence it grows without bound — a factor of 2.9 from 200 points to 1,600. For the golden angle it does not: 1.08 over the same range.

That statistic does not invert, does not need a calibration sample, and is not sensitive to where the rim cut goes. The reason is that it states an asymptotic claim rather than a value at a size: it is about how a quantity behaves as the tiling grows, not about what number it takes on a particular tiling.

The three that misbehave are all values at a size. Area evenness is a coefficient of variation on one point set. Lewis’s slope is a regression on one tiling. Aboav’s $a$ is a regression on one tiling. Each takes a number, and each number’s meaning depends on properties of the sample that the number does not carry.

That is probably the most transferable thing in these two essays. A statistic that compares a tiling to itself at two sizes is much harder to fool than one that reduces a tiling to a value, and where a claim can be posed asymptotically it usually should be.

What is not being claimed #

Not that either relation is wrong. Both are real regularities, both were measured carefully on real material, and both hold on the material they were measured on.

Not that Voronoi tilings are epithelia. They are not: real cells have walls with mechanical properties, divide, and are not the nearest-neighbour regions of their centroids. A Voronoi tiling is a model of an epithelium and a rough one, and every number here inherits that.

What is claimed is narrower and, on a site whose subject arrives pre-loaded with confident statements, is the usual claim: the two relations are quoted together as though they described one kind of object, and measured on two tilings they separate cleanly. Whichever of them a piece of tissue satisfies is information about where that tissue sits on the order–disorder axis, and satisfying both would be the surprising result rather than the expected one.

The one number worth carrying away #

If a single figure is to be remembered from these two essays it should be the one from the last one rather than either fitted constant: in the tiling where Lewis’s law works best, side count explains 32% of the variance in cell area.

A relation that accounts for under a third of the variation is a description of a trend, and the trend is genuine. It is not a constraint on a cell, and every use of Lewis’s law that treats a cell’s side count as determining its size is over-reading it by roughly a factor of three.

That number is not in the original paper, is not in the citations, and takes one line to compute from data anyone fitting the law already has.

The corresponding number for Aboav’s relation on the ordered tiling is better, and it should be said for balance: the topological correlation there is genuinely tight, because the defects are bound pairs and a bound pair is not a statistical tendency but a structure. Where Aboav’s relation holds for that reason, it is describing something real and local.

Which is the position both laws end up in once they are measured rather than cited. Each is a correct description of a specific structural fact about a specific kind of tiling. Neither is a property of tissue, both are quoted as though they were, and the two kinds of tiling they describe are at opposite ends of the only axis that distinguishes them.